The punctured cotangent bundle (excluding the zero section) of the complex projective space of dimension $n$ can be identified with the space:


$$\{(P,Q) \mid P^2=P, tr(P)=1, PQ+QP = Q, \operatorname{tr}(Q) = 0, Q\ne 0\},$$


where $P$ and $Q$ are $(n+1)$ dimensional Hermitian matrices.
Please see Furutani and  Tanaka's article [A Kähler structure on the punctured cotangent bundle of complex and quaternion projective spaces and its application to a geometric quantization I](https://doi.org/10.1215/kjm/1250518882) (Proposition 2.3.).

Sketch of the proof: The space of the matrices $P$ is isomorphic as a
homogeneous space to $CP^n$, since the action of $U(n+1)$ on it is
transitive and the isotropy group of any point is $U(n)$. The conditions
on the matrix $Q$ are obtained by taking the derivatives of the conditions
on $P$, and the identification of $T^*CP^n$ and $TCP^n$ via the
Fubini–Study metric.


  [1]: https://projecteuclid.org/journals/kyoto-journal-of-mathematics/volume-34/issue-4/A-K%C3%A4hler-structure-on-the-punctured-cotangent-bundle-of-complex/10.1215/kjm/1250518882.full